Integrand size = 13, antiderivative size = 94 \[ \int x \sqrt [3]{a+b x^3} \, dx=\frac {1}{3} x^2 \sqrt [3]{a+b x^3}-\frac {a \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} b^{2/3}}-\frac {a \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{6 b^{2/3}} \] Output:
1/3*x^2*(b*x^3+a)^(1/3)-1/9*a*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3 ^(1/2))*3^(1/2)/b^(2/3)-1/6*a*ln(b^(1/3)*x-(b*x^3+a)^(1/3))/b^(2/3)
Time = 0.28 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.50 \[ \int x \sqrt [3]{a+b x^3} \, dx=\frac {6 b^{2/3} x^2 \sqrt [3]{a+b x^3}-2 \sqrt {3} a \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )-2 a \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )+a \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{18 b^{2/3}} \] Input:
Integrate[x*(a + b*x^3)^(1/3),x]
Output:
(6*b^(2/3)*x^2*(a + b*x^3)^(1/3) - 2*Sqrt[3]*a*ArcTan[(Sqrt[3]*b^(1/3)*x)/ (b^(1/3)*x + 2*(a + b*x^3)^(1/3))] - 2*a*Log[-(b^(1/3)*x) + (a + b*x^3)^(1 /3)] + a*Log[b^(2/3)*x^2 + b^(1/3)*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3) ])/(18*b^(2/3))
Time = 0.31 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {811, 853}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \sqrt [3]{a+b x^3} \, dx\) |
\(\Big \downarrow \) 811 |
\(\displaystyle \frac {1}{3} a \int \frac {x}{\left (b x^3+a\right )^{2/3}}dx+\frac {1}{3} x^2 \sqrt [3]{a+b x^3}\) |
\(\Big \downarrow \) 853 |
\(\displaystyle \frac {1}{3} a \left (-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}-\frac {\log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 b^{2/3}}\right )+\frac {1}{3} x^2 \sqrt [3]{a+b x^3}\) |
Input:
Int[x*(a + b*x^3)^(1/3),x]
Output:
(x^2*(a + b*x^3)^(1/3))/3 + (a*(-(ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1 /3))/Sqrt[3]]/(Sqrt[3]*b^(2/3))) - Log[b^(1/3)*x - (a + b*x^3)^(1/3)]/(2*b ^(2/3))))/3
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Sim p[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp [Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]
Time = 0.64 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.29
method | result | size |
pseudoelliptic | \(\frac {6 \left (b \,x^{3}+a \right )^{\frac {1}{3}} x^{2} b^{\frac {2}{3}}+2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right ) a -2 \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right ) a +\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right ) a}{18 b^{\frac {2}{3}}}\) | \(121\) |
Input:
int(x*(b*x^3+a)^(1/3),x,method=_RETURNVERBOSE)
Output:
1/18*(6*(b*x^3+a)^(1/3)*x^2*b^(2/3)+2*3^(1/2)*arctan(1/3*3^(1/2)*(b^(1/3)* x+2*(b*x^3+a)^(1/3))/b^(1/3)/x)*a-2*ln((-b^(1/3)*x+(b*x^3+a)^(1/3))/x)*a+l n((b^(2/3)*x^2+b^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2)*a)/b^(2/3)
Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (71) = 142\).
Time = 0.08 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.02 \[ \int x \sqrt [3]{a+b x^3} \, dx=\frac {6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{2} x^{2} + 6 \, \sqrt {\frac {1}{3}} a b \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (-b^{2}\right )^{\frac {1}{3}} b x - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {2}{3}}\right )} \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}}}{b^{2} x}\right ) - 2 \, \left (-b^{2}\right )^{\frac {2}{3}} a \log \left (-\frac {\left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{x}\right ) + \left (-b^{2}\right )^{\frac {2}{3}} a \log \left (-\frac {\left (-b^{2}\right )^{\frac {1}{3}} b x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{x^{2}}\right )}{18 \, b^{2}} \] Input:
integrate(x*(b*x^3+a)^(1/3),x, algorithm="fricas")
Output:
1/18*(6*(b*x^3 + a)^(1/3)*b^2*x^2 + 6*sqrt(1/3)*a*b*sqrt(-(-b^2)^(1/3))*ar ctan(-sqrt(1/3)*((-b^2)^(1/3)*b*x - 2*(b*x^3 + a)^(1/3)*(-b^2)^(2/3))*sqrt (-(-b^2)^(1/3))/(b^2*x)) - 2*(-b^2)^(2/3)*a*log(-((-b^2)^(2/3)*x - (b*x^3 + a)^(1/3)*b)/x) + (-b^2)^(2/3)*a*log(-((-b^2)^(1/3)*b*x^2 - (b*x^3 + a)^( 1/3)*(-b^2)^(2/3)*x - (b*x^3 + a)^(2/3)*b)/x^2))/b^2
Result contains complex when optimal does not.
Time = 0.72 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.41 \[ \int x \sqrt [3]{a+b x^3} \, dx=\frac {\sqrt [3]{a} x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} \] Input:
integrate(x*(b*x**3+a)**(1/3),x)
Output:
a**(1/3)*x**2*gamma(2/3)*hyper((-1/3, 2/3), (5/3,), b*x**3*exp_polar(I*pi) /a)/(3*gamma(5/3))
Time = 0.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.43 \[ \int x \sqrt [3]{a+b x^3} \, dx=\frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{9 \, b^{\frac {2}{3}}} + \frac {a \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{18 \, b^{\frac {2}{3}}} - \frac {a \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{9 \, b^{\frac {2}{3}}} - \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} a}{3 \, {\left (b - \frac {b x^{3} + a}{x^{3}}\right )} x} \] Input:
integrate(x*(b*x^3+a)^(1/3),x, algorithm="maxima")
Output:
1/9*sqrt(3)*a*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^(1/3) )/b^(2/3) + 1/18*a*log(b^(2/3) + (b*x^3 + a)^(1/3)*b^(1/3)/x + (b*x^3 + a) ^(2/3)/x^2)/b^(2/3) - 1/9*a*log(-b^(1/3) + (b*x^3 + a)^(1/3)/x)/b^(2/3) - 1/3*(b*x^3 + a)^(1/3)*a/((b - (b*x^3 + a)/x^3)*x)
\[ \int x \sqrt [3]{a+b x^3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {1}{3}} x \,d x } \] Input:
integrate(x*(b*x^3+a)^(1/3),x, algorithm="giac")
Output:
integrate((b*x^3 + a)^(1/3)*x, x)
Timed out. \[ \int x \sqrt [3]{a+b x^3} \, dx=\int x\,{\left (b\,x^3+a\right )}^{1/3} \,d x \] Input:
int(x*(a + b*x^3)^(1/3),x)
Output:
int(x*(a + b*x^3)^(1/3), x)
\[ \int x \sqrt [3]{a+b x^3} \, dx=\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} x^{2}}{3}+\frac {\left (\int \frac {x}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}d x \right ) a}{3} \] Input:
int(x*(b*x^3+a)^(1/3),x)
Output:
((a + b*x**3)**(1/3)*x**2 + int(((a + b*x**3)**(1/3)*x)/(a + b*x**3),x)*a) /3